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Abstract <p>Direct scattering problem for the one-dimensional Helmholtz equation is numerically solved. The transfer matrix method and the integral method are used to obtain a second-order accurate implicit difference scheme for the transfer matrix. A duplication strategy, a convolution theorem, and a fast Fourier transform are used to develop an algorithm for accelerated solution of the Helmholtz equation, asymptotically requiring only <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(N\log^{2}N)\)</EquationSource> <!--OptelIns2570084Frumin-m1--> </InlineEquation> arithmetic operations. The scattering problem is numerically solved using the example of an exponential smooth layer, the solution to which is known. The numerical simulation confirmed that the proposed algorithm is accurate and fast, which is necessary in practical applications for optical and acoustic sensing of media in applied optics and acoustics.</p>

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Accelerated Algorithm for Solving the Direct Scattering Problem for the Wave Equation

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • L. L. Frumin,
  • A. E. Chernyavsky

摘要

Abstract

Direct scattering problem for the one-dimensional Helmholtz equation is numerically solved. The transfer matrix method and the integral method are used to obtain a second-order accurate implicit difference scheme for the transfer matrix. A duplication strategy, a convolution theorem, and a fast Fourier transform are used to develop an algorithm for accelerated solution of the Helmholtz equation, asymptotically requiring only \(O(N\log^{2}N)\) arithmetic operations. The scattering problem is numerically solved using the example of an exponential smooth layer, the solution to which is known. The numerical simulation confirmed that the proposed algorithm is accurate and fast, which is necessary in practical applications for optical and acoustic sensing of media in applied optics and acoustics.